Theorem ssrnres 5572

Description: Subset of the range of a restriction. (Contributed by NM, 16-Jan-2006.)

Assertion

Ref	Expression
ssrnres	|- ( B C_ ran ( C |` A ) <-> ran ( C i^i ( A X. B ) ) = B )

Proof of Theorem ssrnres

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		inss2 3834	. . . . 5 |- ( C i^i ( A X. B ) ) C_ ( A X. B )
2		rnss 5354	. . . . 5 |- ( ( C i^i ( A X. B ) ) C_ ( A X. B ) -> ran ( C i^i ( A X. B ) ) C_ ran ( A X. B ) )
3	1, 2	ax-mp 5	. . . 4 |- ran ( C i^i ( A X. B ) ) C_ ran ( A X. B )
4		rnxpss 5566	. . . 4 |- ran ( A X. B ) C_ B
5	3, 4	sstri 3612	. . 3 |- ran ( C i^i ( A X. B ) ) C_ B
6		eqss 3618	. . 3 |- ( ran ( C i^i ( A X. B ) ) = B <-> ( ran ( C i^i ( A X. B ) ) C_ B /\ B C_ ran ( C i^i ( A X. B ) ) ) )
7	5, 6	mpbiran 953	. 2 |- ( ran ( C i^i ( A X. B ) ) = B <-> B C_ ran ( C i^i ( A X. B ) ) )
8		ssid 3624	. . . . . . . 8 |- A C_ A
9		ssv 3625	. . . . . . . 8 |- B C_ _V
10		xpss12 5225	. . . . . . . 8 |- ( ( A C_ A /\ B C_ _V ) -> ( A X. B ) C_ ( A X. _V ) )
11	8, 9, 10	mp2an 708	. . . . . . 7 |- ( A X. B ) C_ ( A X. _V )
12		sslin 3839	. . . . . . 7 |- ( ( A X. B ) C_ ( A X. _V ) -> ( C i^i ( A X. B ) ) C_ ( C i^i ( A X. _V ) ) )
13	11, 12	ax-mp 5	. . . . . 6 |- ( C i^i ( A X. B ) ) C_ ( C i^i ( A X. _V ) )
14		df-res 5126	. . . . . 6 |- ( C |` A ) = ( C i^i ( A X. _V ) )
15	13, 14	sseqtr4i 3638	. . . . 5 |- ( C i^i ( A X. B ) ) C_ ( C |` A )
16		rnss 5354	. . . . 5 |- ( ( C i^i ( A X. B ) ) C_ ( C |` A ) -> ran ( C i^i ( A X. B ) ) C_ ran ( C |` A ) )
17	15, 16	ax-mp 5	. . . 4 |- ran ( C i^i ( A X. B ) ) C_ ran ( C |` A )
18		sstr 3611	. . . 4 |- ( ( B C_ ran ( C i^i ( A X. B ) ) /\ ran ( C i^i ( A X. B ) ) C_ ran ( C |` A ) ) -> B C_ ran ( C |` A ) )
19	17, 18	mpan2 707	. . 3 |- ( B C_ ran ( C i^i ( A X. B ) ) -> B C_ ran ( C |` A ) )
20		ssel 3597	. . . . . . 7 |- ( B C_ ran ( C |` A ) -> ( y e. B -> y e. ran ( C |` A ) ) )
21		vex 3203	. . . . . . . 8 |- y e. _V
22	21	elrn2 5365	. . . . . . 7 |- ( y e. ran ( C |` A ) <-> E. x <. x , y >. e. ( C |` A ) )
23	20, 22	syl6ib 241	. . . . . 6 |- ( B C_ ran ( C |` A ) -> ( y e. B -> E. x <. x , y >. e. ( C |` A ) ) )
24	23	ancrd 577	. . . . 5 |- ( B C_ ran ( C |` A ) -> ( y e. B -> ( E. x <. x , y >. e. ( C |` A ) /\ y e. B ) ) )
25	21	elrn2 5365	. . . . . 6 |- ( y e. ran ( C i^i ( A X. B ) ) <-> E. x <. x , y >. e. ( C i^i ( A X. B ) ) )
26		elin 3796	. . . . . . . 8 |- ( <. x , y >. e. ( C i^i ( A X. B ) ) <-> ( <. x , y >. e. C /\ <. x , y >. e. ( A X. B ) ) )
27		opelxp 5146	. . . . . . . . 9 |- ( <. x , y >. e. ( A X. B ) <-> ( x e. A /\ y e. B ) )
28	27	anbi2i 730	. . . . . . . 8 |- ( ( <. x , y >. e. C /\ <. x , y >. e. ( A X. B ) ) <-> ( <. x , y >. e. C /\ ( x e. A /\ y e. B ) ) )
29	21	opelres 5401	. . . . . . . . . 10 |- ( <. x , y >. e. ( C |` A ) <-> ( <. x , y >. e. C /\ x e. A ) )
30	29	anbi1i 731	. . . . . . . . 9 |- ( ( <. x , y >. e. ( C |` A ) /\ y e. B ) <-> ( ( <. x , y >. e. C /\ x e. A ) /\ y e. B ) )
31		anass 681	. . . . . . . . 9 |- ( ( ( <. x , y >. e. C /\ x e. A ) /\ y e. B ) <-> ( <. x , y >. e. C /\ ( x e. A /\ y e. B ) ) )
32	30, 31	bitr2i 265	. . . . . . . 8 |- ( ( <. x , y >. e. C /\ ( x e. A /\ y e. B ) ) <-> ( <. x , y >. e. ( C |` A ) /\ y e. B ) )
33	26, 28, 32	3bitri 286	. . . . . . 7 |- ( <. x , y >. e. ( C i^i ( A X. B ) ) <-> ( <. x , y >. e. ( C |` A ) /\ y e. B ) )
34	33	exbii 1774	. . . . . 6 |- ( E. x <. x , y >. e. ( C i^i ( A X. B ) ) <-> E. x ( <. x , y >. e. ( C |` A ) /\ y e. B ) )
35		19.41v 1914	. . . . . 6 |- ( E. x ( <. x , y >. e. ( C |` A ) /\ y e. B ) <-> ( E. x <. x , y >. e. ( C |` A ) /\ y e. B ) )
36	25, 34, 35	3bitri 286	. . . . 5 |- ( y e. ran ( C i^i ( A X. B ) ) <-> ( E. x <. x , y >. e. ( C |` A ) /\ y e. B ) )
37	24, 36	syl6ibr 242	. . . 4 |- ( B C_ ran ( C |` A ) -> ( y e. B -> y e. ran ( C i^i ( A X. B ) ) ) )
38	37	ssrdv 3609	. . 3 |- ( B C_ ran ( C |` A ) -> B C_ ran ( C i^i ( A X. B ) ) )
39	19, 38	impbii 199	. 2 |- ( B C_ ran ( C i^i ( A X. B ) ) <-> B C_ ran ( C |` A ) )
40	7, 39	bitr2i 265	1 |- ( B C_ ran ( C |` A ) <-> ran ( C i^i ( A X. B ) ) = B )

Syntax hints:   <-> wb 196   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   _Vcvv 3200   i^i cin 3573   C_ wss 3574   <.cop 4183   X. cxp 5112   ran crn 5115   |` cres 5116

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-xp 5120  df-rel 5121  df-cnv 5122  df-dm 5124  df-rn 5125  df-res 5126

This theorem is referenced by:  rninxp  5573